where \(\omega _k\)  are weights that needs to be determined imposing the
interpolation condition, \(\varphi (\Vert \mathbf {x} - \mathbf {x}_k\Vert )\)  are radial basis
functions where \(\mathbf {x}_k\)  are the coordinates of the active nodes,
\(\lbrace p_i(\mathbf {x})\rbrace _{i=1}^m\)  are monomials that span the space of polynomials with
a specific degree. Adding polynomials to the RBF interpolant functions helps to
properly capture constant and linear features in the given data and ensures positive-definiteness of the RBF function, which in turn implies solvability of
the interpolation problem [1]}. The
coefficient vectors \(\mathbf {\omega } = \left[\omega _1, \dots ,\omega _{N_\text{active}} \right]^T\)  and \(\mathbf {b} = \left[ b_1, \dots , b_m\right]^T\)  are obtained solving the following linear system of equations:
\({\left\lbrace \begin{array}{ll}\left( \mathbf {K} + \sigma ^2 \mathbf {I} \right) \mathbf {\omega }+ \mathbf {P} \mathbf {b} = \mathbf {d}, \\\mathbf {P}^T \mathbf {\omega }= \mathbf {0},\end{array}\right.}\) 
